Marius' reforms did do away with the original unit 'classes'. (for want of a better word)

In the Roman legions, the role of the average soldier was highly dependent upon your income, land ownership and position in society. As a citizen army, each man was responsible for equipping himself to fight, the poorest citizens, and young sons generally constituted the Velites (the most lightly equipped of soldiers, comprised mostly of the 5th social class, the proletarii) Hastati were generally slightly better off, able to afford better equipment, and so were from the higher class. Principes were a step higher, either promoted Hastati, or richer citizens who could afford better equipment. Triarii were usually veterans, who had previously served, rich with spoils from war. Cavalry were generally recruited from the young men of the knight class. (hence Equites - Equestrian order)

So in comes Marius, and the major point of his reforms. By recruiting from the 6th social order, the portion of the citizenry previously referred to as the capite censi (lit. Head Count) potential soldiers could simply not afford to buy their own equipment. Through service they could become enfranchised, Roman Citizens and gain land and spoils. Unsurprisingly many flocked to join the new legions. The state then had to supply the arms and equipment, leading to the creation of the new soldier: the legionary with equipment closely matching the pre-marian Principes. The only way for the state to easily equip and maintain the legions was through standardisation, and Marius accounted for this, his reforms changed how the legion's baggage was handled, their equipment and their organisation as a standing army. Vastly increasing the nascent empire's military effectiveness.

Of course, the reforms were instrumental in the rise of tyrants, the destruction of the republic and the creation of the Roman Empire.

TL;DR It didn't change instantly, but yes, the Marian reforms did remove the previous organisation of the Roman army into Hastati, Principe and Triarii, and in its place put the Legionary.

For further reading look at 'The Making of the Roman Army: from Republic to Empire' Also, Goldsworthy's 'The Roman Army at war 100BC-200AD' for a (lot) more information.